Results 1 to 5 of 5

Math Help - Critical numbers problem

  1. #1
    Junior Member
    Joined
    Oct 2012
    From
    London
    Posts
    54

    Critical numbers problem

    Find all the critical numbers of the function f(x) = x+ln(|cos(x)|) that lie in the interval [pi, 3pi].

    Please help, thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,686
    Thanks
    617

    Re: Critical numbers problem

    Hello, nubshat!

    Exactly where is your difficulty?


    \text{Find all the critical numbers of the function: }\:f(x) \:=\: x\;+\;\ln(|\cos x|)\:\text{ in the interval }[\pi, 3\pi].

    Differentiate and equate to zero: . f'(x) \:=\:1 + \frac{-\sin x}{\cos x} \:=\:0

    We have: . 1 - \tan x \:=\:0 \quad\Rightarrow\quad \tan x \:=\:1

    Hence: . x\:=\:\pm\tfrac{\pi}{4},\:\pm\tfrac{5\pi}{4},\:\pm  \tfrac{9 \pi}{4},\:\pm\tfrac{13\pi}{4}\:\hdots

    Answers: . x \;=\;\frac{5\pi}{4},\:\frac{9\pi}{4}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2010
    Posts
    980
    Thanks
    236

    Re: Critical numbers problem

    Doesn't the derivative fail to exist at \frac{3\pi}{2} and \frac{5\pi}{2}, so that they are critical points, too?

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2012
    From
    London
    Posts
    54

    Re: Critical numbers problem

    would you also have to include the two end points at pi and 3pi?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Mar 2010
    Posts
    980
    Thanks
    236

    Re: Critical numbers problem

    Yes, I think both end points, too. Critical points are only used for finding minima and maxima, so assuming you're finding the minimum or maximum over the range \pi to 3\pi, then you would need to include those points, too.

    I suppose it could be that you're finding the minimum or maximum over a different range, and for some reason you're temporarily limiting yourself to finding critical points in the given range. That doesn't make much sense, though.

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Critical Numbers
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 18th 2010, 01:23 PM
  2. Critical numbers help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 6th 2009, 08:41 AM
  3. Critical Numbers
    Posted in the Calculus Forum
    Replies: 9
    Last Post: November 20th 2009, 07:19 PM
  4. Critical numbers help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 23rd 2009, 11:22 AM
  5. Critical numbers
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 15th 2009, 07:55 AM

Search Tags


/mathhelpforum @mathhelpforum